In the context of modal logics one standardly considers two modal operators: possibility ( $$\Diamond $$ ◊ ) and necessity ( $$\Box $$ □ ) [see for example Chellas (Modal logic. An introduction, Cambridge University Press, Cambridge, 1980)]. If the classical negation is present these operators can be treated as inter-definable. However, negative modalities ( $$\Diamond \lnot $$ ◊ ¬ ) and ( $$\Box \lnot $$ □ ¬ ) are also considered in the literature [see for example Béziau (Log Log Philos 15:99–111, 2006. https://doi.org/10.12775/LLP.2006.006 ); Došen (Publ L’Inst Math, Nouv Sér 35(49):3–14, 1984); Gödel, in: Feferman (ed.), Collected works, vol 1, Publications 1929–1936, Oxford University Press, New York, 1986, p. 300; Lewis and Langford (Symbolic logic, Dover Publications Inc., New York, 1959, p. 497)]. Both of them can be treated as negations. In Béziau (Log Log Philos 15:99–111, 2006. https://doi.org/10.12775/LLP.2006.006 ) a logic $$\mathbf{Z}$$ Z has been defined on the basis of the modal logic $$\mathbf{S5}$$ S 5 . $$\mathbf{Z}$$ Z is proposed as a solution of so-called Jaśkowski’s problem [see also Jaśkowski (Stud Soc Sci Torun 5:57–77, 1948)]. The only negation considered in the language of $$\mathbf{Z}$$ Z is ‘it is not necessary’. It appears that logic $$\mathbf{Z}$$ Z and $$\mathbf{S5}$$ S 5 inter-definable. This initial correspondence result between $$\mathbf{S5}$$ S 5 and $$\mathbf{Z}$$ Z has been generalised for the case of normal logics, in particular soundness-completeness results were obtained [see Marcos (Log Anal 48(189–192):279–300, 2005); Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 34(4):229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski (Log Univ 12:207–219, 2018. https://doi.org/10.1007/s11787-018-0184-9 ) it has been proved that there is a correspondence between $$\mathbf{Z}$$ Z -like logics and regular extensions of the smallest deontic logic. To obtain this result both negative modalities were used. This result has been strengthened in Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 46(3–4):261–280, 2017) since on the basis of classical positive logic it is enough to solely use $$\Box \lnot $$ □ ¬ to equivalently express both positive modalities and negation. Here we strengthen results given in Mruczek-Nasieniewska and Nasieniewski (Log Univ 12:207–219, 2018. https://doi.org/10.1007/s11787-018-0184-9 ) by showing correspondence for the smallest regular logic. In particular we give a syntactic formulation of a logic that corresponds to the smallest regular logic. As a result we characterise all logics that arise from regular logics. From this follows via respective translations a characterisation of a class of logics corresponding to some quasi-regular logics where $$\mathbf{S2}^{\mathbf{0}}$$ S 2 0 is the smallest element. Moreover, if a given quasi-regular logic is characterised by some class of models, the same class can be used to semantically characterise the logic obtained by our translation.

中文翻译：

---

基于正则和拟正则逻辑的标准模态与负模态的对应关系

在模态逻辑的上下文中，人们通常会考虑两个模态运算符：可能性（ $$\Diamond $$ ◊ ）和必要性（ $$\Box $$ □ ）[参见例如 Chelas（模态逻辑。介绍，剑桥大学出版社，剑桥，1980 年）]。如果存在经典否定，则可以将这些运算符视为可相互定义的。然而，文献中也考虑了负面模态 ( $$\Diamond \lnot $$ ◊ ¬ ) 和 ( $$\Box \lnot $$ □ ¬ ) [例如参见 Béziau (Log Log Philos 15:99–111, 2006. https://doi.org/10.12775/LLP.2006.006）；Došen (Publ L'Inst Math, Nouv Sér 35(49):3–14, 1984)；Gödel, in: Feferman (ed.), Collectedworks, vol 1, Publications 1929–1936, Oxford University Press, New York, 1986, p. 300；Lewis 和 Langford（符号逻辑，Dover Publications Inc.，纽约，1959，第 497 页）]。它们都可以被视为否定。在 Béziau (Log Log Philos 15:99–111, 2006. https://doi.org/10.12775/LLP.2006.006 ) 中，基于模态逻辑定义了逻辑 $$\mathbf{Z}$$ Z $$\mathbf{S5}$$ S 5 . $$\mathbf{Z}$$ Z 被提议作为所谓的 Jaśkowski 问题的解决方案 [另见 Jaśkowski (Stud Soc Sci Torun 5:57–77, 1948)]。$$\mathbf{Z}$$ Z 语言中考虑的唯一否定是“没有必要”。看起来逻辑 $$\mathbf{Z}$$ Z 和 $$\mathbf{S5}$$ S 5 是可以相互定义的。$$\mathbf{S5}$$ S 5 和 $$\mathbf{Z}$$ Z 之间的初始对应结果已被推广用于正常逻辑的情况，特别是获得了健全性-完整性结果[参见 Marcos (Log肛门 48(189–192):279–300, 2005)；Mruczek-Nasieniewska 和 Nasieniewski（Bull Sect Log 34(4):229–248, 2005）]。在 Mruczek-Nasieniewska 和 Nasieniewski（Log Univ 12：207–219, 2018. https://doi.org/10.1007/s11787-018-0184-9 ) 已经证明 $$\mathbf{Z}$$ Z 类逻辑和常规扩展之间存在对应关系最小的道义逻辑。为了获得这个结果，使用了两种消极方式。该结果在 Mruczek-Nasieniewska 和 Nasieniewski（Bull Sect Log 46(3–4):261–280, 2017）中得到了加强，因为在经典正逻辑的基础上，仅使用 $$\Box \lnot $$ 就足够了□ ¬ 等效地表达肯定形式和否定形式。在这里，我们通过显示最小正则逻辑的对应关系来加强 Mruczek-Nasieniewska 和 Nasieniewski (Log Univ 12:207–219, 2018. https://doi.org/10.1007/s11787-018-0184-9) 中给出的结果。特别地，我们给出了对应于最小正则逻辑的逻辑的句法公式。因此，我们表征了从常规逻辑产生的所有逻辑。由此通过相应的翻译得出与一些准正则逻辑相对应的一类逻辑的特征，其中 $$\mathbf{S2}^{\mathbf{0}}$$ S 2 0 是最小元素。此外，如果给定的准正则逻辑由某类模型表征，则可以使用同一类从语义上表征通过我们的翻译获得的逻辑。